BERLIN — The euro currency union, a centerpiece of Europe’s efforts to knit its far-flung nations into a coherent whole, edged toward a rupture Thursday when the region’s central bank said it was ready to pull the plug on Cyprus.

The stark ultimatum came in a terse statement Thursday from the European Central Bank’s governing board, which said that on Monday it would cut off the flow of euros to Cyprus’s struggling banks unless the country’s leaders reach agreement with the International Monetary Fund and other European nations on the terms of a $20.5 billion bailout to save their country from financial disaster.

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The deadline sent Cypriot leaders scrambling to find fixes, and by Thursday night they were discussing restructuring the nation’s worst-off bank and imposing capital controls that would sharply restrict depositors’ ability to withdraw money, an effort to prevent bank runs.

Because Cyprus is small and its banks aren’t so wired into the international system, a failure isn’t likely to trigger the kinds of global problems feared if Greece or another euro nation were to leave the currency union.

But a departure of Cyprus from the euro zone would be a stunning blow to the crowning achievement of Europe’s post-World War II effort to unify its warring nations. The euro was intended to be so ironclad that there are no procedures for countries to back out of the currency bloc. But leaders have feared that if one country leaves, pressure could increase on others to follow suit, quickly destabilizing the entire project.

The IMF and other euro-zone countries have offered to lend Cyprus about $13 billion, but they expect the country to come up with $7.5 billion on its own through taxes, government spending cuts or other measures to help restart a banking system that is essentially broke. A plan to raise the money by taxing bank deposits — including tens of billions of dollars held by Russians and other foreigners — collapsed this week in the Cypriot parliament.

The prospect of a Cyprus exit fueled an intense hunt for options — from a nationwide bank restructuring that would put the largest Cypriot banks out of business to more-unusual proposals such as mortgaging the property of the Orthodox Church, selling off natural gas rights or simply asking for donations.

Panicos Demetriades, governor of the Central Bank of Cyprus, called on lawmakers Thursday to vote immediately on a legal framework to rehabilitate the country’s banking sector. The government will also create an Investment Solidarity Fund, which is intended to appeal to “the patriotism of Cypriots” and draw on contributions from ordinary citizens, businessmen and foreign investors.

“We will have a program of support for Cyprus by Monday,” Demetriades said earlier in the day.

Those details, however, were overshadowed by the larger issues — of the ECB, a developed-world central bank, flexing its muscle over a nation’s leaders and of the possibility that the euro zone, after years of insisting otherwise, may finally have to admit that membership to the currency union is not sacrosanct.

Central banks in the developed world have taken on outsize influence since the collapse of Lehman Brothers in 2008. The U.S. Federal Reserve has made massive asset purchases and recently established a target unemployment rate, and the Bank of Japan has committed to engineering higher prices to “reflate” the country’s economy.

Throughout Europe’s three-year-old financial crisis, the ECB has proved an aggressive arbiter in a currency union that remains a political work in progress. At key points, its willingness to venture into uncharted waters by buying government bonds or taking other extraordinary steps has give political leaders time and financial leeway to make tough economic choices — and arguably kept the currency zone intact.

But the ECB has also shown the limits of its patience, most notably when it undercut the government of Italy’s Silvio Berlusconi after the then-leader tried to back out of budget cuts that ECB members felt were important to the country’s financial rehabilitation.

Cyprus’s finances are in such disarray that its banks don’t qualify for the standard ECB loans that euro-zone financial institutions depend on. The alternative, an ECB program called Emergency Liquidity Assistance, is reserved for banks that have a cash-flow crisis but are fundamentally solvent. The ECB has funneled billions of dollars into Cyprus under that program. But with no rescue plan in sight and a massive deposit run likely once the country’s banks reopen, the ECB concluded that it must leave the tiny island to fend for itself.

After meeting in Frankfurt, the board said it would consider lending more to Cyprus only “if an EU/IMF program is in place that would ensure the solvency of the concerned banks.” Cyprus’s bank deposits are more than seven times the size of the local economy. The IMF, in particular, is pushing for a program that will not just keep the country afloat but will dramatically scale back the size of a financial system inflated with tens of billions of dollars from offshore.

European leaders on Thursday were openly discussing the possibility of Cyprus leaving the 17-nation currency union, an unprecedented step. On a practical level, a new Cypriot currency would probably drop sharply in value relative to the euro, wiping out the purchasing power of Cypriots, whose banks would be obliged to do business in the new money. Existing contracts would have to be repriced, touching off complicated rounds of litigation between Cypriot businesses and their international clients and customers.

There could be ramifications for the global economy if a pullout reignited broader concerns about Europe. Nations such as Greece that are struggling to remain in the euro zone’s good graces might see a template for the pros, cons and practicalities of a euro-zone exit. Countries such as Poland that are considering whether to adopt the currency would see powerful evidence of the sweep and authority of the central bank they would be joining.

There are no formal procedures in the euro zone’s treaty for countries to leave or be pushed out of the bloc. But the withdrawal of ECB support would probably be the beginning of the end — leaving the country’s banks without the flow of money needed to operate, isolating them from the world financial system and perhaps leaving Cyprus little choice but to reestablish a national currency.

The Central Bank of Cyprus warned that one large bank, the Cyprus Popular Bank, would probably have to shut its doors Tuesday morning if nothing is done.

Expectations differ on how Europe would be affected if Cyprus fails to come to terms on the bailout. In Germany, where an election-year Parliament will have to vote on any European assistance, public opinion is skeptical of what has been portrayed locally as a bailout of Russian oligarchs. Almost a third of Cyprus’s deposits are estimated to come from Russia, and many European policymakers suspect that the island is a money-laundering hub, making generous rescue packages politically difficult to support.

Given the choice, German leaders widely feel that a Cypriot exit from the euro zone is a risk they are willing to take.

“I think it would be manageable,” said Rainer Bruederle, a candidate for chancellor who is the parliamentary leader of the Free Democrats, the junior member of Germany’s ruling coalition.

But many analysts fear that concerns could quickly spread to larger troubled countries such as Spain and Italy. Italy’s economy is almost 90 times as large as Cyprus’s, making the consequences of what happens on the island of 1.1 million people far graver than if it were to implode in isolation.

Cyprus is “definitely a systemic risk, and the unrest of the last couple of days have proved that, unfortunately,” said Dutch Finance Minister Jeroen Dijsselbloem, the head of the caucus of countries that share the euro, in the European Parliament on Thursday.

Investors have reacted calmly to the turmoil, with markets only modestly down this week. A Thursday bond auction in Spain — a crucial test of whether anxiety about Cyprus was spreading to other vulnerable economies — actually brought lower borrowing costs for bonds with five- and 10-year maturities.

Cypriot Finance Minister Michalis Sarris has been in Moscow since late Tuesday seeking emergency funding from Russian leaders, who have been apoplectic about the possibility that Russian bank deposits in Cyprus would take a hit. Cypriot Energy Minister Georgios Lakkotrypis also was in Moscow, fueling speculation that Cyprus was offering access to its natural gas resources as a sweetener to any deal.

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